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In the étale topology, we have an equivalence of categories between the category of fiber functors on the (small) étale site $Ét(\text{Spec}(S))$ and the category of local strictly henselian $S$-algebras (see for instance the wonderful paper https://arxiv.org/pdf/1407.5782.pdf). The relevance of the fiber functors is that it is sensible to think of them as "Points" in the étale topology.

However, I admit that I don't really have a feel for the local strictly henselian $S$-algebras for any given ring $S$ and would like to see some more examples. For instance, if $S=\mathbb{Q}$, then for each $p$, the algebra $\mathbb{Q}_p^{un}$ is strictly henselian. Are these all the local strictly henselian $\mathbb{Q}$-algebras?

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  • $\begingroup$ The qualification "small" is incorrect. The usual definition of the étale site of a scheme $X$ includes schemes that are not necessarily locally of finite type over $X$, whereas the usual definition of the small (or petit) étale site excludes schemes that are not étale over $X$. Either way, these are not equivalent to the site considered in the linked paper. As for examples of strictly henselian local algebras, any separably closed field is an example (albeit one where the maximal ideal is zero). $\endgroup$
    – Zhen Lin
    Commented Sep 3, 2020 at 2:56
  • $\begingroup$ To be clear, you know that you can take the strict Henselization of any local ring, right? Are you asking whether these are the strictly local rings of $\mathrm{Spec}(\mathbb{Z})$ (or at least the generic points of such)? $\endgroup$ Commented Sep 3, 2020 at 10:31
  • $\begingroup$ @ZhenLin Thank you for the correction! I'll edit the question accordingly. $\endgroup$ Commented Sep 3, 2020 at 12:23
  • $\begingroup$ @AlexYoucis I would like to know the local strict $\mathbb{Q}$-algebras. I know of the strict henselization, but I'm wondering if we can obtain all local strict $\mathbb{Q}$-algebras that way. $\endgroup$ Commented Sep 3, 2020 at 12:25
  • $\begingroup$ But, if you take any local $\mathbb{Q}$-algebra, you can take its strict Henselization and obtain a strictly Henselian local $\mathbb{Q}$-algebra, right? $\endgroup$ Commented Sep 3, 2020 at 12:31

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There are other local strictly henselian $\mathbb{Q}$-algebras, for example the algebraic closure $\overline{\mathbb{Q}}$, the complex numbers $\mathbb{C}$ or the ring of power series $\mathbb{C}[[t]]$.

In this context, a local strictly henselian $\mathbb{Q}$-algebra is precisely a strictly henselian local ring $R$ equipped with a ring morphism $\mathbb{Q}\to R$. This ring morphism is necessarily unique, so you can rephrase the condition by saying that $R$ should contain $\mathbb{Q}$. Fields are local rings, and they are stricly henselian if and only if they are separably closed. So any separably closed field of characteristic 0 is a local strictly henselian $\mathbb{Q}$-algebra.

The étale site studied in the wonderful article by Gabber and Kelly would be called the "big étale site". The problem is that there are two possible definitions of big étale site. The one used by Gabber—Kelly takes as objects of the site only schemes that are of finite presentation over the base scheme $S$. This has the advantage that the topos of sheaves on the site is a Grothendieck topos. For this Grothendieck topos, you can compute the category of points: it is equivalent to the category of local strictly henselian $\mathbb{Q}$-algebras (and ring morphisms between them).

There is another site that is often called the "big étale site". Here you take as objects of the site all schemes over the base scheme $S$ (or all affine schemes, or something similar). This has the advantage that the category of sheaves on it contains a full subcategory equivalent to the category of schemes over $S$ (via the Yoneda Lemma). A disadvantage is that the category of sheaves is no longer a Grothendieck topos. It might still make sense to compute the category of points, but it will be very complicated.

Then there is also the small étale site: the sheaves on this site form a Grothendieck topos, and the points for this topos are the strictly henselian local rings $R$ that can be written as a filtered colimit $R = \varinjlim_{i \in I} R_i$, where each $R_i$ is an $R$-algebra with $R \to R_i$ étale. In particular, $\overline{\mathbb{Q}}$ is a point for the small étale topos over $\mathbb{Q}$, and it is the unique point up to isomorphism.

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